Subgroups — C₂×C₅₃×#2231 #27241148

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2231	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#365	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#360	C₂	1	2	yes	maximal
7	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#131×C₅₃	—	2	1	no	—
9	C₂×#125×C₅₃	—	2	1	no	—
10	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#499	—	2	2	no	—
12	C₅₃×#499	C₂²	1	4	yes	complemented
13	C₅₃×#443	—	2	2	no	—
14	C₅₃×#443	C₂²	1	4	yes	—
15	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#107×C₅₃	—	2	1	no	—
17	C₂×#106×C₅₃	C₂²	1	2	yes	—
18	C₂×#104×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#365	C₂²	1	8	yes	complemented
21	C₅₃×#364	C₂²	1	4	yes	complemented
22	C₅₃×#360	C₂²	1	8	yes	—
23	C₂×#41×C₅₃	—	2	1	no	—
24	#135×C₅₃	—	2	1	no	—
25	#135×C₅₃	—	2	1	no	—
26	#135×C₅₃	Q₈	1	2	yes	—
27	#135×C₅₃	C₂³	1	2	yes	—
28	C₂×Q₁₆×C₅₃	Q₈	1	1	yes	characteristic
29	C₂×SD₁₆×C₅₃	—	2	1	no	—
30	C₂×D₈×C₅₃	Q₈	1	1	yes	characteristic
31	#131×C₅₃	—	2	4	no	—
32	#131×C₅₃	—	4	2	no	—
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₂×M₁₆×C₅₃	—	4	1	no	—
35	C₂²×C₈×C₅₃	—	4	1	no	—
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	#125×C₅₃	—	2	4	no	—
39	#125×C₅₃	—	4	2	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	M₃₂×C₅₃	—	2	2	no	—
44	M₃₂×C₅₃	C₂³	1	4	yes	—
45	C₂×C₁₆×C₅₃	—	2	2	no	—
46	#108×C₅₃	—	2	1	no	—
47	#108×C₅₃	C₂³	1	2	yes	—
48	#107×C₅₃	—	2	4	no	—
49	#106×C₅₃	—	2	2	no	—
50	#106×C₅₃	C₂³	1	4	yes	—
51	#104×C₅₃	—	2	4	no	—
52	C₄×C₈×C₅₃	D₄	1	2	yes	—
53	C₄×C₈×C₅₃	C₂³	1	2	yes	—
54	#41×C₅₃	—	2	2	no	—
55	#41×C₅₃	—	2	2	no	—
56	#41×C₅₃	—	4	1	no	—
57	#41×C₅₃	—	4	1	no	—
58	C₂×Q₈×C₅₃	—	2	1	no	—
59	C₂×D₄×C₅₃	—	2	1	no	—
60	C₂×D₄×C₅₃	—	4	1	no	—
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	—	4	2	no	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₅₃	—	2	1	no	—
66	Q₁₆×C₅₃	C₂×Q₈	1	2	yes	—
67	SD₁₆×C₅₃	—	2	2	no	—
68	SD₁₆×C₅₃	—	2	2	no	—
69	D₈×C₅₃	—	2	1	no	—
70	D₈×C₅₃	C₂×Q₈	1	2	yes	—
71	M₁₆×C₅₃	—	2	2	no	—
72	M₁₆×C₅₃	—	4	1	no	—
73	M₁₆×C₅₃	—	4	4	no	—
74	C₂×C₈×C₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	—	4	4	no	—
76	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
77	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
78	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
79	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	4	8	no	—
88	C₄⋊C₄×C₅₃	—	4	2	no	—
89	C₄²×C₅₃	—	2	1	no	—
90	C₄²×C₅₃	C₂×D₄	1	2	yes	—
91	C₁₆×C₅₃	—	2	4	no	—
92	C₂³×C₅₃	—	4	1	no	—
93	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
94	Q₈×C₅₃	—	2	2	no	—
95	Q₈×C₅₃	—	4	1	no	—
96	D₄×C₅₃	—	2	2	no	—
97	D₄×C₅₃	—	4	2	no	—
98	D₄×C₅₃	—	4	2	no	—
99	D₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	1	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	8	4	no	—
107	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₅₃	—	4	2	no	—
114	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
117	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
118	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
119	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
120	C₂×#2231	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	4	2	no	—
123	C₂²×C₅₃	—	8	2	no	—
124	C₂²×C₅₃	#476	1	1	yes	characteristic
125	C₂²×C₅₃	#476	1	1	yes	characteristic
126	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₄×C₅₃	—	4	2	no	—
128	C₄×C₅₃	—	4	2	no	—
129	C₄×C₅₃	—	8	4	no	—
130	C₄×C₅₃	#476	1	1	yes	characteristic
131	C₄×C₅₃	#476	1	1	yes	characteristic
132	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
133	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
134	C₂×#499	C₂×C₅₃	1	1	yes	characteristic, complemented
135	C₂×#443	C₂×C₅₃	1	1	yes	characteristic
136	#2231	C₂×C₅₃	1	8	yes	direct product factor
137	C₂×#365	C₂×C₅₃	1	2	yes	complemented
138	C₂×#364	C₂×C₅₃	1	1	yes	characteristic, complemented
139	C₂×#360	C₂×C₅₃	1	2	yes	—
140	C₂×C₅₃	—	2	1	no	—
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	8	2	no	—
143	C₂×C₅₃	#2231	1	2	yes	direct product factor
144	C₂×C₅₃	C₂×#476	1	1	yes	characteristic
145	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
146	C₂×#131	—	2	1	no	—
147	C₂×#125	—	2	1	no	—
148	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
149	#499	—	2	2	no	—
150	#499	C₂²×C₅₃	1	4	yes	complemented
151	#443	—	2	2	no	—
152	#443	C₂²×C₅₃	1	4	yes	—
153	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
154	C₂×#107	—	2	1	no	—
155	C₂×#106	C₂²×C₅₃	1	2	yes	—
156	C₂×#104	—	2	1	no	—
157	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
158	#365	C₂²×C₅₃	1	8	yes	complemented
159	#364	C₂²×C₅₃	1	4	yes	complemented
160	#360	C₂²×C₅₃	1	8	yes	—
161	C₅₃	C₂×#2231	1	1	yes	characteristic, direct product factor, Sylow
162	C₂×#41	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	—	2	1	no	—
165	#135	Q₈×C₅₃	1	2	yes	—
166	#135	C₂³×C₅₃	1	2	yes	—
167	C₂×Q₁₆	Q₈×C₅₃	1	1	yes	characteristic
168	C₂×SD₁₆	—	2	1	no	—
169	C₂×D₈	Q₈×C₅₃	1	1	yes	characteristic
170	#131	—	2	4	no	—
171	#131	—	4	2	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	4	1	no	—
174	C₂²×C₈	—	4	1	no	—
175	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
177	#125	—	2	4	no	—
178	#125	—	4	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	4	2	no	—
181	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
182	M₃₂	—	2	2	no	—
183	M₃₂	C₂³×C₅₃	1	4	yes	—
184	C₂×C₁₆	—	2	2	no	—
185	#108	—	2	1	no	—
186	#108	C₂³×C₅₃	1	2	yes	—
187	#107	—	2	4	no	—
188	#106	—	2	2	no	—
189	#106	C₂³×C₅₃	1	4	yes	—
190	#104	—	2	4	no	—
191	C₄×C₈	D₄×C₅₃	1	2	yes	—
192	C₄×C₈	C₂³×C₅₃	1	2	yes	—
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	4	1	no	—
196	#41	—	4	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
204	Q₁₆	—	2	1	no	—
205	Q₁₆	C₂×Q₈×C₅₃	1	2	yes	—
206	SD₁₆	—	2	2	no	—
207	SD₁₆	—	2	2	no	—
208	D₈	—	2	1	no	—
209	D₈	C₂×Q₈×C₅₃	1	2	yes	—
210	M₁₆	—	2	2	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	4	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
216	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
217	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
220	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
221	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
222	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
223	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
224	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	8	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	C₂×D₄×C₅₃	1	2	yes	—
230	C₁₆	—	2	4	no	—
231	C₂³	—	4	1	no	—
232	C₂³	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
233	Q₈	—	2	2	no	—
234	Q₈	—	4	1	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
247	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
249	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
250	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
254	C₈	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
255	C₈	C₂×SD₁₆×C₅₃	1	2	yes	—
256	C₈	C₂×SD₁₆×C₅₃	1	2	yes	—
257	C₈	C₂²×Q₈×C₅₃	1	1	yes	characteristic
258	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	C₅₃×#476	1	1	yes	characteristic
263	C₂²	C₅₃×#476	1	1	yes	characteristic
264	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	8	4	no	—
268	C₄	C₅₃×#476	1	1	yes	characteristic
269	C₄	C₅₃×#476	1	1	yes	characteristic
270	C₄	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
271	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	8	2	no	—
275	C₂	C₅₃×#2231	1	2	yes	direct product factor
276	C₂	C₂×C₅₃×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₅₃×#499	C₂	semidirect
3. C₅₃×#2231	C₂	direct (N × H)
4. C₂×C₅₃×#365	C₂	semidirect
5. C₂×C₅₃×#364	C₂	semidirect
12. C₅₃×#499	C₂²	semidirect
20. C₅₃×#365	C₂²	semidirect
21. C₅₃×#364	C₂²	semidirect
120. C₂×#2231	C₅₃	direct (N × H)
134. C₂×#499	C₂×C₅₃	semidirect
136. #2231	C₂×C₅₃	direct (N × H)
137. C₂×#365	C₂×C₅₃	semidirect
138. C₂×#364	C₂×C₅₃	semidirect
143. C₂×C₅₃	#2231	direct (N × H)
150. #499	C₂²×C₅₃	semidirect
158. #365	C₂²×C₅₃	semidirect
159. #364	C₂²×C₅₃	semidirect
161. C₅₃	C₂×#2231	direct (N × H)
275. C₂	C₅₃×#2231	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅₃×#499	2
C₂×C₅₃×#443	2
C₅₃×#2231	2
C₂×C₅₃×#365	2
C₂×C₅₃×#364	2
C₂×C₅₃×#360	2
C₂×#2231	53

Subgroups — C2×C53×#2231 #27241148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2231 #27241148